Heuristic exploration of the relationship between cases and viral load
Marlin Lee
12/24/2021
This report looks at exploring the relationship between wastewater and cases. There are four components to this analysis.
Removing putative outliers
Binning analysis
Smoothing signal
Statistical analysis
This report does not present any final answers but presents some very convincing heuristics.
1 Data: The first look
The two data sets used in this analysis are the Madison case data sourced from the Wisconsin DHS and wastewater concentration data produced by the Wisconsin State Laboratory of Hygiene. This wastewater data has entries every couple of days from November 15 2020 to June 24 2021.
| Date | Site | Cases | SevenDayMACases | N1 | BCoV | N2 | PMMoV | GeoMeanN12 | FlowRate | temperature | equiv_sewage_amt |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 2021-01-25 | Madison | 53 | 99.85714 | 135003 | 11.44000 | 233165 | 31732106 | 177420.3 | 37.28 | NA | 1 |
| 2021-01-26 | Madison | 60 | 98.85714 | 1831813 | 6.93000 | 1686654 | 29748189 | 1757735.7 | 37.74 | NA | 1 |
| 2021-01-27 | Madison | 177 | 97.71429 | 1357329 | 2.05900 | 1323010 | 30102158 | 1340059.6 | 36.17 | NA | 1 |
| 2021-01-28 | Madison | 111 | 94.28571 | 231625 | 2.39800 | 265869 | 41230863 | 248157.0 | 37.07 | NA | 1 |
| 2021-01-31 | Madison | 57 | 95.42857 | 256082 | 3.75643 | 233472 | 26460001 | 244515.8 | 36.30 | NA | 1 |
| 2021-02-01 | Madison | 46 | 94.14286 | 149707 | 1.86200 | 108584 | 27005372 | 127498.2 | 36.28 | NA | 1 |
The case data has a strong weekend effect so for this section we look at a seven day smoothing of cases. The simple display of the data shows the core components of this story. First that wastewater data is very noisy. And that there is a hint of a relationship between the two signals.
FirstImpressionDF <- FullDF%>%
NoNa(SelectedIndVar,"Cases")
FirstImpression <- FirstImpressionDF%>%#Removing outliers
ggplot(aes(x=Date))+#Data depends on time
geom_point(aes(y=(Cases), color="Cases",info=Cases),size = 1)+
geom_line(aes(y=MinMaxFixing(!!sym(SelectedIndVar),Cases),
color=SelectedIndVar,
info=!!sym(SelectedIndVar)))+#compares SelectedIndVar to Cases
geom_line(aes(y=(SevenDayMACases),
color="Seven Day MA Cases",
info=Cases))+
labs(y="Reported cases")+
ColorRule
ggplotly(FirstImpression)%>%
add_lines(x=~Date, y=FirstImpressionDF[[SelectedIndVar]],
yaxis="y2", data=FirstImpressionDF, showlegend=FALSE, inherit=FALSE) %>%
layout(yaxis2 = SecondAxisFormat,
legend=list(title=list(text=''),x = 1.15, y = 0.9),
margin = list(l = 50, r = 75, b = 50, t = 50, pad = 4))#To remoive weekend effects we are looking at the 7 day smoothing of cases.2 Removing potential outliers
Looking at the wastewater measurements we observe minimal movement in N1 until it spikes up to expected levels. This is a sign that N1 before 11/20/2020 might have failed to capture the true amount of Covid shed in the community. To get the best picture between N1 and cases we removed these points from the analysis. In addition there were some points many times larger than adjacent values hinting at them being outliers. For the same reason we removed these points from the analysis.
DaySmoothed=21#Very wide smoothing to find where the data strong deviate from trend
ErrorMarkedDF <- FullDF%>%#Remove older Var data that clearly has no relationship to Cases
mutate(FlagedOutliers = IdentifyOutliers(!!sym(SelectedIndVar), Action = "Flag"),
ReplacedOutliers = ifelse(FlagedOutliers,NA,!!sym(SelectedIndVar)),
ReplacedOutliers = ifelse(Date == mdy("01/26/2021") | Date == mdy("01/27/2021"), NA, ReplacedOutliers))
#Calculating error Limits
#EarlyOutlier | LargeError | LargeError2| LargeError3
ErrorRemovedDF <- ErrorMarkedDF%>%
mutate(!!OutlierName := ifelse(FlagedOutliers,!!sym(SelectedIndVar),NA))%>%
mutate(!!SelectedIndVar := ReplacedOutliers)
#!is.na(Cases),
#SevenDayMACases
Ref <- ErrorRemovedDF%>%
filter(!(is.na(!!sym(SelectedIndVar))&is.na(!!sym(OutlierName))))%>%
ggplot(aes(x=Date))+#Data depends on time
geom_line(aes(y=!!sym(SelectedIndVar),
color=SelectedIndVar,
info = !!sym(SelectedIndVar)))+#compares Var to Cases
geom_point(aes(y=!!sym(OutlierName),
color=OutlierName,
info = !!sym(OutlierName)))+
ColorRule
ggplotly(Ref,tooltip=c("info","Date"))%>%
layout(yaxis = SecondAxisFormat,
legend=list(title=list(text=''),x = 1.15, y = 0.9),
margin = list(l = 50, r = 75, b = 50, t = 50, pad = 4))UpdatedDF <- ErrorRemovedDF3 Binning
To isolate this relationship we used a primitive binning relationship. We used non overlapping bins of 2 weeks and took the median of the data within that range. This reduces autocorrelation issues and masks potential noise in the data. We see a very strong trend once the potential outliers are removed.
#StartDate is Where the binning starts
#DaySmoothing is The size of the bins
#Lag is The offset between Cases and Var
BinnedVarName <- paste("Binned",SelectedIndVar)
Bining <- function(DF,StartDate=1,DaySmoothing=14,Lag=0){
BinDF <- DF%>%
select(Date, Cases, !!sym(SelectedIndVar))%>%
mutate("Moved Cases" = data.table::shift(Cases, Lag),#moving SLD lag days backwards
Week=(as.numeric(Date)+StartDate)%/%DaySmoothing)%>%#putting variables into bins via integer division
group_by(Week)%>%
#filter(Week>2670)%>%
summarise("Binned Cases" := median(!!sym("Moved Cases"), na.rm=TRUE),
!!BinnedVarName := (median((!!sym(SelectedIndVar)),
na.rm=TRUE)),
Date = mean(Date,na.rm = TRUE))#summarize data within bins.
return(BinDF)
}
BinErrorRemovedDF <- Bining(UpdatedDF)
BinErrorKeptDF <- Bining(FullDF)
DiffrenceDF <- inner_join(BinErrorRemovedDF,BinErrorKeptDF,by=c("Week","Date"))%>%
filter(!!paste0(BinnedVarName,".x") != !!paste0(BinnedVarName,".y"))
BinedCorrGraph <- ggplot()+
geom_segment(aes(x = !!sym("Binned Cases.x"),
y = !!sym(paste0(BinnedVarName,".x")),
xend = !!sym("Binned Cases.y"),
yend = !!sym(paste0(BinnedVarName,".y"))),
data = DiffrenceDF)+
geom_point(aes(x = !!sym("Binned Cases"),
y = !!sym(BinnedVarName),
color = "outliers not removed",
info = Date),
size = 2,
data = BinErrorKeptDF,
shape=15)+
geom_point(aes(x = !!sym("Binned Cases"),
y = !!sym(BinnedVarName),
color = "outliers removed",
info = Date),
data = BinErrorRemovedDF)+
ggtitle(paste(BinnedVarName,",Cases removed potential outliers"))+
#geom_abline(slope = 3000)+
labs(x="Binned Cases",y=BinnedVarName)
ggplotly(BinedCorrGraph,tooltip=c("x","y","info"))%>%
layout(legend=list(title=list(text=''),x = 1.15, y = 0.9),
margin = list(l = 50, r = 75, b = 50, t = 50, pad = 4))#cor(BinDF$BinnedN1, BinDF$BinnedCases, use="pairwise.complete.obs")
#summary(lm(BinnedCases~BinnedN1, data=BinDF))
OutputBinning <- data.frame(row.names=c("correlation"),
WithOutliers = c(cor(BinErrorKeptDF[[BinnedVarName]],
BinErrorKeptDF[["Binned Cases"]],
use="pairwise.complete.obs")),
WithOutOutliers = c(cor(BinErrorRemovedDF[[BinnedVarName]],
BinErrorRemovedDF[["Binned Cases"]],
use="pairwise.complete.obs")))
formattable(OutputBinning)| WithOutliers | WithOutOutliers | |
|---|---|---|
| correlation | 0.9040724 | 0.9130309 |
4 Data smoothing
The goal in this section is to smooth the data to get a similar effect without losing resolution.
4.1 Case smoothing
A key component to this is that the relationship between N1 and Case involves a gamma distribution modeling both the time between catching Covid-19 and getting a test and the concentration of the shedded particles. We found a gamma distribution with mean 11.73 days and a standard deviation of 7.68 gives good results and match’s other research (Fernandez-Cassi et al. 2021).
Mean <- 11.73
StandardDeviation <- 7.68
Scale = StandardDeviation^2/Mean
Shape = Mean/Scale
SLDWidth <- 21
weights <- dgamma(1:SLDWidth, scale = Scale, shape = Shape)
par(mar=c(4,4,4,10))
plot(weights,
main=paste("Gamma Distribution with mean =",Mean, "days,and SD =",StandardDeviation),
ylab = "Weight",
xlab = "Lag")
SLDSmoothedDF <- UpdatedDF%>%
mutate(
SLDCases = c(rep(NA,SLDWidth-1),#elimination of starting values not relevant as we have a 50+ day buffer of case data
rollapply(Cases,width=SLDWidth,FUN=weighted.mean,
w=weights,
na.rm = FALSE)
#,rep(NA,10)
))#no missing data to remove
#CasesMinMaxPrep = MinMaxNormalization(SLDSmoothedDF$SLDCases,Prep=TRUE)
SLDPlot = SLDSmoothedDF%>%
NoNa("SLDCases")%>%#same plot as earlier but with the SLD smoothing
ggplot(aes(x=Date))+
geom_line(aes(y=Cases,
color="Cases" , info = Cases),alpha=.2)+
geom_line(aes(y=SevenDayMACases,
color="Seven Day MA Cases" , info = SevenDayMACases),alpha=.4)+
geom_line(aes(y=SLDCases, color="SLD Cases",info = SLDCases))+
labs(y="Reported Cases")+
ColorRule
ggplotly(SLDPlot,tooltip=c("info","Date"))%>%
layout(legend=list(title=list(text=''),x = 1.15, y = 0.9),
margin = list(l = 50, r = 75, b = 50, t = 50, pad = 4))5 Towards a formal analysis
Cross correlation and Granger Causality are key components to formalize this analysis. Cross correlation looks at the correlation at a range of time shifts and Granger analysis performs a test for predictive power. These results must be kept in mind that the data removal was not done rigorously and the resulting data set has aspects of being non stationary. Therefore these statistics are good to keep in mind they should not be used for concrete results.
CCFChar <- function(ccfObject){
LargestC = max(ccfObject$acf)
Lag = which.max(ccfObject$acf)-21
return(c(LargestC,Lag))
}
ModelTesting <- function(DF,Var1,Var2){
UsedDF <- DF%>%
NoNa(Var1,Var2)#removing rows from before both series started
Vec1 <- unname(unlist(UsedDF[Var1]))
Vec2 <- unname(unlist(UsedDF[Var2]))
CCFReport <- CCFChar(ccf(Vec1,Vec2,na.action=na.pass,plot = FALSE))
VarPredCase <- grangertest(Vec1, Vec2, order = 1)$"Pr(>F)"[2]
CasePredVar <- grangertest(Vec2,Vec1, order = 1)$"Pr(>F)"[2]
return(round(c(CCFReport,CasePredVar,VarPredCase),4))
}
#ErrorRemovedDF
BaseLine <- ModelTesting(FullDF,SelectedIndVar,"Cases")
BaseLineSevenDay <- ModelTesting(FullDF,SelectedIndVar,"SevenDayMACases")
ErrorRemoved <- ModelTesting(UpdatedDF,SelectedIndVar,"SevenDayMACases")
SLDVar <- ModelTesting(SLDSmoothedDF,SelectedIndVar,"SLDCases")
SevenLoess <- ModelTesting(SLDSmoothedDF,loessVar,"SevenDayMACases")
SLDLoess <- ModelTesting(SLDSmoothedDF,loessVar,"SLDCases")
Output <- data.frame(row.names=c("Max Cross Correlation","Lag of largest Cross correlation","P-value WasteWater predicts Cases","P-value Cases predicts wastewater"),
CasesvsVar = BaseLine,
SevenDayMACasesvsVar = BaseLineSevenDay,
ErrorRemoved = ErrorRemoved,
SLDVar = SLDVar,
SevenLoess = SevenLoess,
SLDLoess = SLDLoess)
OutputRightPosition <- transpose(Output)
colnames(OutputRightPosition) <- rownames(Output)
rownames(OutputRightPosition) <- c(paste("Section 1: Cases vs" , SelectedIndVar),
paste("Section 1: 7 Day MA Cases vs" , SelectedIndVar),
paste("Section 2: Cases vs" , SelectedIndVar),
paste(" Section 4.1: SLD Cases vs ",SelectedIndVar),
paste("Section 4.2: 7 Day MA Cases vs Loess smoothing of ",SelectedIndVar),
paste("Section 4.2: SLD Cases vs Loess smoothing of ",SelectedIndVar))
formattable(OutputRightPosition)| Max Cross Correlation | Lag of largest Cross correlation | P-value WasteWater predicts Cases | P-value Cases predicts wastewater | |
|---|---|---|---|---|
| Section 1: Cases vs N1 | 0.6137 | 1 | 0.0000 | 0.0000 |
| Section 1: 7 Day MA Cases vs N1 | 0.7066 | 1 | 0.0000 | 0.0002 |
| Section 2: Cases vs N1 | 0.8275 | 1 | 0.0000 | 0.0000 |
| Section 4.1: SLD Cases vs N1 | 0.8107 | 1 | 0.0000 | 0.1902 |
| Section 4.2: 7 Day MA Cases vs Loess smoothing of N1 | 0.9263 | 1 | 0.0113 | 0.0000 |
| Section 4.2: SLD Cases vs Loess smoothing of N1 | 0.8834 | 1 | 0.0000 | 0.0000 |